
\(f^{\prime }(R)<0\)  makes the scalar degree of freedom appearing in the theory a ghost. To eradicate the possibility of a ghost degree of freedom, one must require \(f^{\prime }(R)>0\)  for all \(R\)  (or at least within the domain under consideration). We will assume it to hold throughout our consideration.

\(f^{\prime \prime }(R)<0\)  is related to unstable growth of curvature perturbations in the weak gravity limit (This is also known as the tachyonic instability or the Dolgov-Kawasaki instability [1]}). Therefore, one requires that \(f^{\prime \prime }(R)>0\)  at least around the matter domination epoch.

